Playing in his fourth game of the season, Neal -- a former 40-goal scorer -- netted his first of the season late in the first period to give the Penguins a lead they didn't relinquish against the road-weary Predators.
Nashville lost the final four of a franchise-record, 17-day, seven-game trip. Gabriel Bourque scored 2:28 in but the Predators were held to fewer than two goals for the fourth consecutive game.
The Penguins have held opponents under three goals in seven of eight games.
Nashville fell to 7-1-2 when scoring first.
Bourque, who scored his second of the season, missed Nashville's previous game because of the birth of his first child.
Kevin Klein's shot from the point deflected off of Malkin's stick and Penguins defenseman Olli Maatta's skate before Bourque shoved the puck Fleury.
Fleury has allowed two or fewer goals in nine of his 10 home starts this season.
That lead lasted just 10 minutes before Dupuis and Neal scored 2:20 apart to put the Penguins in front in the first period.
Dupuis' second of the season -- a rising slap shot from the top of the left circle off a set faceoff play -- came at 12:36, shortly after a Nashville killed a penalty for too many men on the ice.
But with the Predators' Ryan Ellis of for interference minutes later, Neal got a piece of Malkin's slap shot from the point while he stood in front of rookie goalie Marek Mazanec.
The 22-year-old Mazanec made his NHL debut a week earlier and had his first career start Tuesday. He stopped 21 of 25 shots in two periods, and fellow rookie Carter Hutton played the third.
Mazanec made a spectacular glove save on Neal early in the second period, but Letang made it 3-1 with his fourth of the season at 7:58.
After skipping the morning skate because of illness, Letang scored off a feed from Malkin. Sutter made it 4-1 with 1:23 left in the second.
About a minute later, Nashville's Eric Nystrom was knocked out of the game by a punch to the head during a fight with Tanner Glass.
Bloodied from a cut on his forehead, Nystrom needed assistance and wobbled to the dressing room. He didn't return.
The previous time Nystrom played against the Penguins on Feb. 29, 2012, he delivered a controversial hit to the head of Letang, and the Pittsburgh defenseman was out of the lineup for two weeks. Then with the Dallas Stars, Nystrom was given a roughing penalty but wasn't disciplined further by the league. Glass wasn't with the Penguins then.
Pirates manager Clint Hurdle received a long ovation when he was recognized on the video board for being named the National League manager of the year this week. . Penguins D Paul Martin returned after missing two games because of a lower-body injury. Predators C Mike Fisher also returned from a lower-body ailment, playing for the first time in five games. . The game was the 600th for Penguins LW Chris Kunitz and the 200th for Pittsburgh D Deryk Engelland. . Kunitz had gone more than 19 months since last being held consecutive games without a point. He was pointless for the third straight game.
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PITTSBURGH, PA - NOVEMBER 15: Matt Cullen #7 of the Nashville Predators skates for the loose puck against Olli...